Decomposition of Iterative Equation Solver

• Look for concurrency in loop iterations
  • In this case iterations are really dependent
  • Iteration (i, j) depends on iterations (i, j-1) and (i-1, j)
  • Each anti-diagonal can be computed in parallel
  • Must synchronize after each anti-diagonal (or pt-to-pt)
  • Alternative: red-black ordering (different update pattern)
• Can update all red points first, synchronize globally with a barrier and then update all black points
  • May converge faster or slower compared to sequential program
  • Converged equilibrium may also be different if there are multiple solutions
  • Ocean simulation uses this decomposition
• We will ignore the loop-carried dependence and go ahead with a straight-forward loop decomposition
  • Allow updates to all points in parallel
  • This is yet another different update order and may affect convergence
  • Update to a point may or may not see the new updates to the nearest neighbors (this parallel algorithm is non-deterministic)

  while (!done)    diff = 0.0;    for_all i = 0 to n-1       for_all j = 0 to n-1          temp = A[i, j];          A[i, j] = 0.2(A[i, j]+A[i, j+1]+A[i, j-1]+A[i-1, j]+A[i+1, j]);          diff += fabs (A[i, j] – temp);       end for_all    end for_all    if (diff/(n*n) < TOL) then done = 1; end while

• Offers concurrency across elements: degree of concurrency is n²
• Make the j loop sequential to have row-wise decomposition: degree n concurrency